#250845

order	512
nilpotency class	3
isoclinism	isoclinic class of #250775
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2403
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#61843	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1029 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 32 points:

(2,8)(3,10)(4,13)(5,6)(12,14)(15,16)(17,18,23,19)(20,28,30,32)(21,24,22,26)(25,31,27,29);(1,2)(3,11)(4,7)(5,13)(6,8)(9,12)(10,15)(14,16)(18,24)(19,26)(21,22)(25,28)(27,32)(29,31);(1,3,7,12)(2,9,4,11)(5,10,6,14)(8,15,13,16)(17,19,23,18)(20,27,30,25)(21,26,22,24)(28,31,32,29);(1,4)(2,7)(3,11)(5,8)(6,13)(9,12)(10,15)(14,16)(17,20,23,30)(18,25,19,27)(21,31,22,29)(24,28,26,32)

Polycyclic presentation:

2^9|a^2=i,c^2=h,d^2=i,e^2=h,f^2=hi,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fhi,g^a=gi,c^b=cg,e^b=eh,g^b=gi,e^c=ei,f^c=fhi,g^c=gi,e^d=eh,f^d=fi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 158041);